Galileo's Paradox and the Project of the Reals


Galileo's Paradox and the Project of the Reals 30/12/06


This post consists of two parts. The first is a reprise of the
Galileo Paradox of about a month ago. There are three reasons for this.
Firstly, I have had time to consider set density measures. This is
certainly something I should have explicitly considered beforehand.
Wikipedia, for example, presents the Schnirelmann measure as satisfying our
intuitions about some denumerable sets being bigger than others.
Secondly, the standard of argument against my finished position in
that thread was rather shoddy. I am not knocking the group, or any
partcular contributors to it. On the contrary, the discussion in that
thread was an invaluable help to me in clarifying my ideas. Ultimately,
though, the arguments were disappointing. A clearer restatement of my
position might help.
The third reason brings us to the second part of the post, and it
concerns the consequences of the argument from Galileo's Paradox. At first
I was inclined to think these were not that serious. There still seemed
something slightly odd about a sizeless (countable) infinity, and a
super-infinity which is bigger than that. But I was prepared to believe
that bijection would as it were be the new size, the yardstick with which
to compare infinite sets, that normal or pre-systematic notions of size
would recede into insignificance. It seems to me now that the usual proofs
do not carry through when the notion of (countable) infinity is understood
correctly.

Galileo's Paradox:

When one compares the (infinite) sequence of squares with the
sequence of naturals, there seems, on the one hand, to be more naturals
than square; if however one compares the sequence of successive squares
with their root naturals, there seems on the other hand to be exactly as
many squares as naturals.
The intuitions involved here are I believe very simple and strictly
similar. Firstly, one compares finite sequences or successively larger
finite sequences of naturals with the squares and non-squares of which it
is composed, and observes that this compostion relationship goes on
forever. Secondly, one considers the simple bijection between squares and
roots for finite sequences and observes that this goes on forever.
Obviously it is the presumption of the paradox that the notions of
'more' and 'fewer' involved are unambiguous and perspicuous. Critics should
note that my position is that this is correct, that there is a way of
resolving the paradox which does not involve disambiguation of notions of
size, and that indeed this is the plausible and correct way to resolve the
paradox. It is therefore wholly preposterous to insist that I define my
terms ('more', 'fewer') in advance. If you wish to resolve the paradox by
disambiguating the notion of size, all well and good. But it is not enough
just to wave notions of set density around. You have to carry the argument
through and demonstrate convincingly that multiple notions of size are
actually invoked in the naive intuitions of the paradox. You should
especially beware of ending up with notions of size whose sole raison detre
is the elimination of the paradox in question, which is about the worst
kind of intellectual stupidity one is likely to meet. Set density (and all
density measures add is quantification) may certainly be relevant as the
way to compare different denumerable sets, but that does not mean there is
any confusion of size because of it, and the argument that this is so is so
far quite lacking.

Essentially, the conclusion I would draw from Galileo's Paradox is
that size does not apply to infinite sets. (Countably infinite, let us say
for now.)
The conclusion from the first naive intuition (that |N| >
|Squares|) should be the subtly weaker one that N cannot be said to be the
same size as S. The implication of the second intuition should be,
likewise, not that |N| = |S|, but the weaker conclusion that N cannot be
said to be larger than S. In this sense both intuitions are satisfied, but
the conclusion is that size cannot be an attribute of infinite sets. The
sets go on forever. That is the intuition put in, and the intuition that
comes out but purified of any notion of size.

I spoke of countably infinte sets being elastic. The justification
for this was that we could add or subtract elements from N and still
produce denumerable sets. The metaphor should not be stretched too far (if
you will forgive the pun). A bijection between, for example the Naturals
and the Evens, means that the Evens can be encoded by the Naturals. So is:
1, 2, 3, ........
understood simply as N, different from:
1, 2, 3, ....
understood as code for the Evens, different? No. (This is adapted from an
argument by William Hughes in another thread.) N is not even the same size
as itself. This is not really bizarre. It's simply a consequence of the
fact that size does not apply to denumerable sets.


Let me turn now to the consequences, specifically to Cantor's
proofs. The diagonal argument first. It is convenient to suppose the reals
in the putative list are expressed in binary base, so that we can refer
simply to the inverse of the diagonal.
It is tempting to argue straight off that since we cannot say that
the table of reals is square then the diagonal argument doesn't work. But
this would be shallow at best,. The real point is not that it might just as
well be rectangular (longer than it is wide) as square, but that it does
not have any shape at all, because denumerable has no size but just goes on
forever. We will assume only that there is a bijection between the reals
and naturals, which is all we need to assume for Cantor's argument.
Firstly, one should notice that the move:

for all n, the inverse of diagonal differs from the nth entry

can be mirrored by the move:

for all n, the inverse of the diagonal is not an example of a real not on
the list.

For what can we mean by 'all n' but all finite n? So one could as well
argue from the second move that there is no real not on the list as argue
from the first move that the diagonal is a real not on the list.

This might be fancy footwork, and what one needs is some guiding
idea. And this, from our encounter with GP is that the construction of the
diagonal can never be finished. If we get away from the idea (the
assumption, the grip of the idea) that size still exists for infinite sets,
that the countably infinite lies somewhere in the middle of a range with
finite entities at one end and super-duper infinities at the other, then
Cantor's inverse inverse-diagonal is no real (leaving aside the question of
the existence of irrationals in general), it is the pot of gold at the end
of the rainbow, it is the fantasy of a real, it is merely the project for a
real.

Consider a reformulation of the diagonal argument. This time we
will consider the diagonal itself, moving diagonally across as entries are
added to the table, and the inverse of the diagonal considered as an entry
in the table with its horizontal numbers. In this form the argument will be
that the diagonal and horizontal lines must intersect. (Actually this
version will make a useful version when we come to the power set argument.)
But where they intersect it is impossible for the inverse relation to hold
-- the value must be either 1 or 0. Now of course Cantor makes no
assumption, does not need to make any assumption, about the nature or
structure of the bijection between N and R. Nor will I. I do not even
believe there is such a bijection. What I doubt is whether that can be
proved. But if there were such a bijection, is it not entirely possible
that the postion of the diagonal and its inverse in the list were somehow
related structurally, so that for example, just as a wild possibility,
after n entries in the list, the position of the diagonal so created so far
was always n + 1 in the list? Why not? But then there would be no
intersection. Whatever the merits of that particular argument, the lesson
from GP is that the defining characteristic of infinity is a simple
never-endingness that cannot be quantified or quantified over, and this
means that the sought after intersection is fundamentally elusive.

The power set argument tells us, essentially, that assuming a
bijection between a set of elements and its power set, the subset
consisting of elements not included in their corresponding subsets cannot
be formed. What we need is some way of presenting this argument which makes
apparent that we are dealing with infinite sets. We can write a
characteristic function table with the elements at the top and their
corresponding subsets down the side, so that a subset is expressed as a
horizontal line of 1s and 0s denoting which elements are included in that
set. The diagonal is then in effect the set of elements (wherever there is
a 1 the element is included) which are included in their corresponding
subset by the bijection. The inverse of the diagonal can be considered as
an entry hoizontally which will intersect the diagonal, setting up the
contradiction. This is the identical picture to the one presented in the
above paragraph.

I don't know what the real consequences of all this are, other than
consigning transfinite cardinals to the rubbish dump. Could the whole
concept of the reals be somehow suspect, despite the fact that there
clearly are irrationals? At least it does not seem to be something which
set theorists are uniquely placed to comment on.

Eckard Blumschein, if you are about or lurking, I would be grateful
for your thoughts, since I have arrived at a position which is quite close
to yours, I think.

On closing, I'm suddenly aware that some of the argumentation may
have been rather dense. I am willing to expand if any one requires it. And
of course I will do my best to acknowledge anyone who is willing and able
to demonstrate any errors or dispute debatable points.

Six Letters




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